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Consider the following homogeneous linear DE: y" + 8a y' + 4 y(t) = 0 where a is a constant. Indicate for what intervals of a the system is undamped, under- damped, critically damped, and over-damped. In each case, (i) give a qualitative description of the oscillations, (ii) give an expression for the roots of the characteristic equation in terms of a – simplify as much as possible, and state if the roots are real, or imaginary (complex). iii) write the general solution for undamped, under-damped, critically damped, and over- damped (in terms of a if necessary).

Consider the following homogeneous linear DE: y" + 8a y' + 4 y(t) = 0 where a is a constant. Indicate for what intervals of a the system is undamped, under- damped, critically damped, and over-damped. In each case, (i) give a qualitative description of the oscillations, (ii) give an expression for the roots of the characteristic equation in terms of a – simplify as much as possible, and state if the roots are real, or imaginary (complex). iii) write the general solution for undamped, under-damped, critically damped, and over- damped (in terms of a if necessary).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following homogeneous linear DE: y" + 8 a? y' + 4 y(t) = 0 where a is a constant. Indicate for what intervals of a the system is undamped, under- damped, critically damped, and over-damped. In each case, (i) give a qualitative description of the oscillations, (ii) give an expression for the roots of the characteristic equation in terms of a – simplify as much as possible, and state if the roots are real, or imaginary (complex). (iii) write the general solution for undamped, under-damped, critically damped, and over- damped (in terms of a if necessary).
Transcribed Image Text:Consider the following homogeneous linear DE: y" + 8 a? y' + 4 y(t) = 0 where a is a constant. Indicate for what intervals of a the system is undamped, under- damped, critically damped, and over-damped. In each case, (i) give a qualitative description of the oscillations, (ii) give an expression for the roots of the characteristic equation in terms of a – simplify as much as possible, and state if the roots are real, or imaginary (complex). (iii) write the general solution for undamped, under-damped, critically damped, and over- damped (in terms of a if necessary).
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